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Chapter 2 Determining Position and Velocity Vectors of Earth and Mars for a Given Date .. 4. Position ...... Print. 10. Prussing, John E., and Bruce A. Conway.
THE PENNSYLVANIA STATE UNIVERSITY SCHREYER HONORS COLLEGE

DEPARTMENT OF AEROSPACE ENGINEERING

SURVEY OF EARTH-MARS TRAJECTORIES USING LAMBERT’S PROBLEM AND APPLICATIONS

DAVIDE CONTE Spring 2014

A thesis submitted in partial fulfillment of the requirements for baccalaureate degrees in Aerospace Engineering and Mathematics with honors in Aerospace Engineering

Reviewed and approved* by the following: David B. Spencer Professor of Aerospace Engineering Thesis Supervisor Robert G. Melton Professor of Aerospace Engineering Director of Undergraduate Studies George A. Lesieutre Professor of Aerospace Engineering Head of the Department of Aerospace Engineering Honors Adviser * Signatures are on file in the Schreyer Honors College.

i

ABSTRACT The focus of this thesis is to analyze interplanetary transfer maneuvers from Earth to Mars by solving Lambert’s Problem. Additionally, the orbital parameters of the transfer orbits as well as the relative required ΔVs and times of flights were determined in order to define the optimal departure and arrival windows for a given range of date. The first step in solving Lambert’s Problem consists in finding the positions and velocities of the departure (Earth) and arrival (Mars) planets for a given range of dates. Then, by solving Lambert’s problem for various combinations of departure and arrival dates, porkchop plots can be created and examined. Some of the key parameters that are plotted on porkchop plots and used to investigate possible transfer orbits are the departure characteristic energy, C3, and the arrival v∞. These parameters, in combination with given desired initial and final parking orbital conditions about Earth and Mars, were also used to determine the total ΔV for the various Earth-Mars transfers. Lastly, ΔV results were used to find the necessary amount of propellant needed for the transfer maneuvers as a percentage of total spacecraft mass for a given specific impulse. Moreover, this thesis also considers cases when Lambert’s solution fails to give reasonable results, particularly when the transfer angle between Earth and Mars is close to 0 or 180 degrees. A special case of Lambert’s problem is the Hohmann transfer, which is also discussed in this thesis. The results of the analysis lead to the choice of departure and arrival dates that, given the capabilities of the launch system such as ΔV, makes the mission feasible. Additionally, unless specified otherwise, only time of flight that would be reasonable for an unmanned mission to Mars were considered.

ii

TABLE OF CONTENTS List of Figures .......................................................................................................................... iv List of Tables ........................................................................................................................... vi Acknowledgements .................................................................................................................. vii Chapter 1 Introduction ............................................................................................................. 1 Assumptions ..................................................................................................................... 2 MATLAB ......................................................................................................................... 3 Lambert’s Problem ........................................................................................................... 3 Chapter 2 Determining Position and Velocity Vectors of Earth and Mars for a Given Date .. 4 Position Vectors of Earth and Mars ................................................................................. 4 Velocity Vectors of Earth and Mars................................................................................. 7 Chapter 3 Lambert’s Problem .................................................................................................. 10 Derivation of Lambert’s Problem Solution Using the Universal Variable ...................... 12 Minimum Energy Solution............................................................................................... 14 Algorithm for Solving Lambert’s Problem ...................................................................... 15 Planetary Departure.......................................................................................................... 16 Planetary Arrival .............................................................................................................. 18 Sphere of Influence Considerations ................................................................................. 19 Total ΔV Calculation and Propellant Mass ...................................................................... 20 Chapter 4 Porkchop Plots......................................................................................................... 22 Key Features of a Porkchop Plot ...................................................................................... 24 ΔV vs. Departure Date vs. Arrival Date........................................................................... 26 Propellant Mass ................................................................................................................ 28 Chapter 5 Orbital Parameters of the Transfer Trajectories and Examples............................... 30 Eccentricity ...................................................................................................................... 30 Semimajor Axis................................................................................................................ 33 Inclination ........................................................................................................................ 35 Trajectory Examples ........................................................................................................ 39 Extreme Cases .................................................................................................................. 44 Chapter 6 A Particular Case and Applications......................................................................... 50 The Hohmann Transfer .................................................................................................... 50 Application of Lambert’s Problem to Fast Trajectories ................................................... 51

iii Conclusion and Future Work ................................................................................................... 61 Appendix A Lambert Solution – Flowchart .................................................................... 63 Appendix B Creating Porkchop Plots – Flowchart ......................................................... 64 Appendix C Trajectory Details ....................................................................................... 66 BIBLIOGRAPHY ............................................................................................................ 68

iv

LIST OF FIGURES Figure 1 – Planetary orbit in heliocentric ecliptic frame; in the figure, R and V are respectively the position and the velocity vectors of the planet taken into consideration. Source: Curtis, p.471.7 .............................................................................. 5 Figure 2 – Position and velocity relative to the pqw perifocal reference frame. Source: Curtis, p. 114.5.................................................................................................................. 7 Figure 3 – Geometry of Lambert problem for an interplanetary trajectory. Source: Kemble, p. 9.9 ................................................................................................................... 10 Figure 4 – Departure orbit geometry. Source: Curtis, p. 443.7 ................................................ 16 Figure 5 – Planetary approach. Source: Curtis, p. 451.7 .......................................................... 18 Figure 6 – Porkchop plot for the years 2020-2022 timeframe. The orange dot represents the sample point used in this section to explain how to read a porkchop plot. ................ 23 Figure 7 – Example of a Type I transfer with launch on September 8, 2020 and arrival on December 17, 2020. ......................................................................................................... 25 Figure 8 – Example of a Type II transfer with launch on June 1, 2020 and arrival on May 16, 2021............................................................................................................................ 26 Figure 9 – ΔVtot for the 2020-2022 Earth-Mars transfer trajectories. ...................................... 27 Figure 10 – Propellant mass as a percentage of total initial mass for the 2020-2022 timeframe. ........................................................................................................................ 29 Figure 11 – Example of Earth-Mars trajectories with e < 1..................................................... 32 Figure 12 – Example of Earth-Mars trajectories including e > 1. ............................................ 33 Figure 13 – Example of various semimajor axes for Earth-Mars trajectories ......................... 34 Figure 14 – Inclination of the transfer orbital plane for various Earth-Mars trajectories ........ 36 Figure 15 – An example of an Earth-Mars trajectory with relatively high inclination. ........... 37 Figure 16 – Example of an extremely high inclination orbital transfer from Earth to Mars.... 38 Figure 17 – Mariner 4 trajectory. ............................................................................................. 40 Figure 18 – MRO trajectory. .................................................................................................... 41 Figure 19 – Mars Science Laboratory trajectory. .................................................................... 42 Figure 20 – MAVEN trajectory. .............................................................................................. 43

v Figure 21 – Hyperbolic Earth-Mars trajectory case 1. ............................................................. 46 Figure 22 – Hyperbolic Earth-Mars trajectory case 2. ............................................................. 47 Figure 23 – Hyperbolic Earth-Mars trajectory case 3. ............................................................. 48 Figure 24 – Hyperbolic Earth-Mars trajectory case 4. ............................................................. 49 Figure 25 – Contour plot of trajectory efficiency η centered about the 2029 opposition. ....... 54 Figure 26 – Contour plot of trajectory efficiency η centered about the 2033 opposition. ....... 55 Figure 27 – Contour plot of trajectory efficiency η centered about the 2035 opposition. ....... 56 Figure 28 – Earth-Mars optimized fast transfer for the year 2028-2029. ................................ 58 Figure 29 – Earth-Mars optimized fast transfer for the year 2033. .......................................... 59 Figure 30 – Earth-Mars optimized fast transfer for the year 2035. .......................................... 60

Figure A 1 – Flowchart showing the main steps when solving Lambert’s problem. ............... 63

Figure B 1 – Flowchart showing the main steps when creating porkchop plots. ..................... 65

vi

LIST OF TABLES Table 1 – Comparison between position vectors of Earth and Mars using Standish’s algorithm and using JPL HORIZONS; note that the z-component is considerably smaller than the x and y components. .............................................................................. 6 Table 2 - Comparison between velocity vectors of Earth and Mars using Standish’s algorithm and using JPL HORIZONS; note that the z-component is considerably smaller than the x and y components. .............................................................................. 9 Table 3 – Key results of the Earth-Mars trajectory that minimizes ΔVtot in the 2020-2022 timeframe. ........................................................................................................................ 28 Table 4 – Summary of trajectories presented in this section. .................................................. 39 Table 5 – Summary of some examples of hyperbolic trajectories with low transfer times. .... 44 Table 6 – Example of a Hohmann transfer from Earth to Mars............................................... 51 Table 7 – Earth-Mars oppositions and launch opportunities, 2014-2036. Launch opportunities and arrival dates assume a Hohmann transfer.[15]....................................... 53 Table 8 – Earth-Mars fast transfers with highest efficiency for TOFmax = 150 days and Isp = 1000 seconds. ........................................................................................................ 57

Table C 1 – Earth-Mars trajectories of spacecraft sent to Mars in the past and MAVEN. All of the parameters were computed using MATLAB scripts derived from those used to compute porkchop plots....................................................................................... 66 Table C 2 – Earth-Mars trajectories of hypothetical future missions. All of the parameters were computed using MATLAB scripts derived from those used to compute porkchop plots. ................................................................................................................. 67

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ACKNOWLEDGEMENTS I would like to thank Dr. Spencer for all of the help he gave me in writing this thesis, and especially for teaching Interplanetary Astrodynamics, one of the most exciting classes I have taken at Penn State. I would also like to acknowledge Dr. Melton and Dr. Lesieutre for the teachings and advices they have given me throughout my college career. Additionally, I would like to thank Dr. Parsons who has given me the mathematical foundations and the support to succeed in my later years at Penn State.

Since I was a child, my dad would take me outside on our balcony and show me the splendors of the universe through the lenses of our telescope. Soon, the love for space my family had given me combined with the passion for science, mathematics, and engineering made me realize that I wanted to make a difference in the future of space exploration. Everything I have worked on, especially during the past five years, was made possible by my parents, Giovanni and Rossana, and my sister, Lucia. Their love and financial support have given me the opportunity to study thousands of miles away from home in one of the best universities in the world. And while I occasionally doubted my own abilities, they never did. Lucia, mamma e papà, grazie di cuore per tutto ciò che avete fatto per me, per gli aiuti morali ed economici che mi avete dato, per i consigli che ho ricevuto nei momenti difficili e per la gioia che mi date quando ci sentiamo e vediamo su internet e soprattutto per la gioia che mi date quando torno a casa in Italia. Anche se forse vi verrà difficile capire alcune delle sue parti, vi dedico questa tesi.

I also would like to thank Sour Roberta, my elementary school teacher, who has supported me with her prayers and thoughts.

viii A special thanks goes to the many friends I have made in America since I started studying abroad. Dedico questa tesi anche ai miei amici italiani. Grazie per il vostro supporto e per la vostra ineguagliabile amicizia.

Finally, I would like to thank all of the engineers, scientists, and technicians who work towards space exploration and towards answering the fundamental questions of our seemingly infinite Universe. Their work is an inspiration for me and gives me hope that seeing humans walk on the surface of Mars and other celestial bodies will soon become a reality.

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Chapter 1 Introduction Humanity has always had the desire to explore. In fact, exploration is indeed found within the innate sense of curiosity and enthusiasm that mankind possesses. From religious meaning, depicted as the angry god of war, to astronomical interest because of its strange and apparent retrograde motion, Mars, also known as the red planet, is still considered a remote destination, yet to be reached by humans. In order to fully understand and answer some of the fundamental questions about the formation of our Solar System and the development of life on our planet, interplanetary space exploration has gradually become a necessity. Due to the recent budget cuts, space mission cost has become a more important key factor in deciding whether a mission will take place. In fact, the cost of unmanned space missions to Mars ranges from a few hundreds millions to billions of USD. For example, the Mars Science Laboratory (MSL) was $2.5 billion USD.8 Additionally, traditional chemical propulsion systems having a specific impulse (Isp) of 300-350 seconds require 80-85% of the total spacecraft mass to be propellant, allowing only for relatively small payloads to be delivered to Mars. Therefore, it is clear that it is necessary to make propellant-efficient decisions starting from the orbital trajectory design. Optimal trajectories will not only reduce the cost of missions to Mars, but they will also allow for more massive payloads to be transported in orbit around and possibly land on the red planet. This thesis presents a method of determining the interplanetary transfer orbit parameters, the required ΔVs to achieve the Earth-Mars trajectory, and the payload to propellant mass ratio

2 for given user defined conditions including: departure and arrival dates, Earth and Mars parking orbits altitudes, and specific impulse.

Assumptions The assumptions that were used throughout the analysis of the transfer maneuvers are as follows: 1. All the ΔV maneuvers are treated as impulsive; assuming the use of chemical or nuclear propulsion systems, the burn times required for such ΔVs (in the order of minutes or hours) are considerably much less than the orbital transfer times from Earth to Mars (in the order of months). 2. The only major forces that were considered for the analysis are the point mass gravitational forces of Earth, Mars and the Sun within their respective spheres of influence. Perturbations such as solar radiation pressure, non-uniform gravitational fields, and gravitational effects of other bodies, such as the other planets of the Solar System, were neglected. 3. The method of patched conics was used. This method assumes that when a spacecraft is outside the sphere of influence of a planet, it follows an unperturbed Keplerian orbit (i.e. twobody dynamics) around the Sun. Since interplanetary distances are so vast, for heliocentric orbits it is reasonable to neglect the size of the spheres of influence of Earth and Mars, and consider them to be just points in space coinciding with the planetary centers. Additionally, within each planet’s sphere of influence, the spacecraft travels an unperturbed Keplerian path about the planet. While the sphere of influence appears small on the scale of the solar system, from the point of view of the planet it is “large enough” to be considered to extend to infinity.

3 MATLAB MATLAB® is a high-level language and interactive environment for numerical computation, visualization, and programming. 4 MATLAB was used in this research to implement the necessary algorithms into computer codes as shown in the flowcharts of Appendix A and Appendix B. All the graphical results shown in this thesis were obtained by running the appropriate MATLAB scripts.

Lambert’s Problem

Most of the analysis conducted in this thesis is based on the formulation and solution of a problem in the field of astrodynamics that the Swiss mathematician Johann Heinrich Lambert (1728 – 1779) formulated in 176110. This problem is now known as Lambert’s problem. In spacecraft targeting applications, it is essential to know the time of flight required to travel a particular trajectory. In fact, as discussed in greater details in Chapter3, Lambert’s problem is a boundary value problem involving two position vectors and a time of flight between them. Additionally, according to Lambert’s conjecture, only these three parameters, two position vectors and time of flight, are necessary to uniquely define an orbital transfer between the two points. From the solution to Lambert’s problem it is then possible to calculate the required ΔVs. In the case of an Earth-Mars transfer, the first step required to set up Lambert’s problem is being able to identify Earth’s and Mars’ positions for given departure and arrival dates. Chapter 2 discusses how to obtain the position and velocity vectors of the two planets as a function of time.

4

Chapter 2 Determining Position and Velocity Vectors of Earth and Mars for a Given Date In order to analyze possible transfer orbits from Earth to Mars it is necessary to obtain the position and velocity vectors of the planets at the desired departure and arrival dates. Various method to do so exist, but in this thesis only two main methods are considered: Standish’s algorithm12 and JPL HORIZONS14. Note that the orbital position and velocity of Earth and Mars are described by three-dimensional vectors although the transfer orbit between the two planets is a two-dimensional transfer. In fact, the orbital transfer defines a plane on which the Sun, the Earth at departure and Mars at arrival lie on.

Position Vectors of Earth and Mars Determining the position and velocity vectors of Earth and Mars can be done using the results presented in Standish. Here, a detailed algorithm of how to determine the position of the eight planets (Mercury through Neptune) and Pluto for a given date is offered. Note that such dates must be inputted as Julian Ephemeris Dates. To convert dates from the everyday day/month/year format to Julian Dates, Algorithm 14 taken from Vallado, p.18913. Additionally, it must be noted that Standish’s algorithm provides approximate positions for the planets. This method should not be used unless its errors are acceptable for the application taken into account. High precision ephemerides for the planets are available via the JPL HORIZONS system. The analysis and the following results presented in this thesis are based on the accuracy of Standish’s algorithm, although a comparison with the HORIZONS system was also done.

5 Once the position vectors of Earth and Mars are obtained, it is important to make sure that both vectors are expressed in the same inertial reference frame. For simplicity and convenience, the J2000 ecliptic plane was used. By definition, this plane contains both the Sun and Earth. Additionally, this reference frame is defined by having the x-axis aligned with Earth’s vernal equinox, the z-axis pointing normal to the ecliptic plane, while the y-axis completes the right-handed coordinate system as shown in Figure 1. Moreover, the z-position of Earth with respect to the Sun is exactly zero on January 1, 2000. Standish’s algorithm makes use of the J2000 ecliptic plane as described above. The reference frame in which HORIZONS works can be chosen from a set of predefined reference frames, and it was set to the J2000 ecliptic reference frame for the following applications.

Figure 1 – Planetary orbit in heliocentric ecliptic frame; in the figure, R and V are respectively the position and the velocity vectors of the planet taken into consideration. Source: Curtis, p.471.7

6 For example, consider an arbitrary date, say January 22, 2014 at 00:00:00.0 (UT). The position vectors of Earth and Mars in J2000 ecliptic reference frame are given as shown in Table 1. Comparing the results obtained by using Standish’s algorithm with those of JPL HORIZONS, it can be noted that the difference is relatively small. In fact, for the purpose of the analysis done in this thesis, the results obtained using Standish’s algorithm are sufficient. Additionally, note that the position vector of a planet can be expressed as: ̂ 𝑟⃑ 𝐸 = 𝑟𝑥 𝐼̂ + 𝑟𝑦 𝐽̂ + 𝑟𝑧 𝐾

(1)

̂ are the unit vectors that define the J2000 ecliptic coordinate frame, ‘E’. where 𝐼̂, 𝐽̂, 𝐾 Table 1 – Comparison between position vectors of Earth and Mars using Standish’s algorithm and using JPL HORIZONS; note that the z-component is considerably smaller than the x and y components.

Earth’s Position (AU) on January 22, 2014 at 00:00:00.0 (UT) 𝑟𝑥

𝑟𝑦

𝑟𝑧

Standish’s algorithm

-0.51686

0.83751

-0.000026827

JPL HORIZONS

-0.51570

0.83531

-0.000120055

Mars’ Position (AU) on January 22, 2014 at 00:00:00.0 (UT) 𝑟𝑥

𝑟𝑦

𝑟𝑧

Standish’s algorithm

-1.60258

0.44615

0.048689

JPL HORIZONS

-1.60145

0.44411

0.048592

Additionally, using either method for calculating the position vectors of the planets, the eccentric anomaly can be found. Note that both methods assume that, unlike for the standard twobody problem, the orbital elements of the planets’ orbits change in time. Even though such changes are minimal and become more noticeable in an interval of time of the order of centuries, they were still taken into account in the analysis that follows.

7 Velocity Vectors of Earth and Mars Once the position vectors of the planets are found, it is necessary to compute the velocity vectors. First, velocity vectors in perifocal coordinates are calculated. As Figure 2 shows, the perifocal coordinate reference frame is defined with the center being the focus (in this case the Sun); by the p-coordinate pointing in the direction of the planet’s periapse location, by the wcoordinate pointing normal to the orbital plane such that the planet’s motion is counterclockwise and by the q-coordinate which completes the right-handed reference frame. Additionally, x̅ and y̅ are the components of the distance of the planet from the Sun in the p-direction and q-direction respectively; hence, by Pythagorean Theorem, r = √x̅ 2 + y̅ 2 is the magnitude of the distance.

Figure 2 – Position and velocity relative to the pqw perifocal reference frame. Source: Curtis, p. 114.7

8 Successively, a direction cosine matrix can be used to obtain the velocity vectors in J2000 coordinate frame. In perifocal coordinates, the velocity vector of a planet orbiting the Sun is defined as follows: 𝜇𝑆𝑢𝑛 sin 𝜃 𝑝

𝑣⃑ 𝑃 = −√

𝜇𝑆𝑢𝑛 (𝑒 𝑝

𝑝̂ + √

+ cos 𝜃)𝑞̂

(2)

where 𝑝 is semi-latus rectum, 𝜃 is true anomaly, 𝑒 is eccentricity, and 𝜇𝑆𝑢𝑛 is the gravitational parameter of the Sun. While 𝜇𝑆𝑢𝑛 and 𝑒 are known, 𝑝 and 𝜃 must be calculated. Given eccentric anomaly 𝐸, which was obtained through Standish’s algorithm, and knowing semi-major axis 𝑎, true anomaly 𝜃 and semi-latus rectum 𝑝 can be calculated: 1+𝑒

𝐸

𝜃 = 2 tan−1 [√1−𝑒 tan ( 2 ) ]

(3)

𝑝 = 𝑎 (1 − 𝑒 2 )

(4)

The direction cosine matrix (DCM) used to convert 𝑣⃑ 𝑃 into 𝑣⃑ 𝐸 such that 𝑣⃑ 𝐸 = 𝐶 𝑃𝐸 𝑣⃑ 𝑃 is given by: cos Ω cos 𝜔 − sin Ω sin 𝜔 cos 𝑖 𝐶 𝑃𝐸 = [sin Ω cos 𝜔 + cos Ω sin 𝜔 cos 𝑖 sin 𝜔 sin 𝑖

− cos Ω sin 𝜔 − sin Ω cos 𝜔 cos 𝑖 − sin Ω sin 𝜔 + cos Ω cos 𝜔 cos 𝑖 cos 𝜔 sin 𝑖

sin Ω sin 𝑖 − cos Ω sin 𝑖 ] cos 𝑖

(5)

Note that once the perifocal frame is determined, Eq. (5) is the DCM used to convert from perifocal to J2000 ecliptic reference frame. Therefore, the velocity vectors can now be rewritten in the J2000 ecliptic reference frame as: ̂ 𝑣⃑ 𝐸 = 𝑣𝑥 𝐼̂ + 𝑣𝑦 𝐽̂ + 𝑣𝑧 𝐾 Table 2 shows a comparison between the velocity x,y, and z components obtained using Standish’s algorithm and those obtained from JPL HORIZONS.

(6)

9

Table 2 - Comparison between velocity vectors of Earth and Mars using Standish’s algorithm and using JPL HORIZONS; note that the z-component is considerably smaller than the x and y components.

Earth’s Velocity (km/s) on January 22, 2014 at 00:00:00.0 (UT) 𝑣𝑥

𝑣𝑦

𝑣𝑧

Standish’s algorithm

-25.8345

-15.7563

-0.00050470

JPL HORIZONS

-25.8265

-15.7390

-0.00066392

Mars’ Velocity (km/s) on January 22, 2014 at 00:00:00.0 (UT) 𝑣𝑥

𝑣𝑦

𝑣𝑧

Standish’s algorithm

-5.59172

-21.2728

-0.50470

JPL HORIZONS

-5.58394

-21.2665

-0.30861

Given a departure date from Earth and an arrival date at Mars, it is the possible to determine what the planets’ position and velocities would be on those days. This information will be used to solve Lambert’s problem as explained in the following chapter.

10

Chapter 3 Lambert’s Problem In order to determine the transfer trajectory orbital parameters and the necessary ΔVs to accomplish such transfer, it is essential to solve Lambert’s Problem. In astrodynamics, Lambert’s problem is a two-point boundary value problem (see Figure 3) for the following governing equation of motion: 𝜇𝒓

𝒓̈ = − 𝑟3

(7)

In the case of an Earth to Mars transfer, the boundary conditions become: 𝑟⃗1 (𝑡1 ) = 𝑟⃗(𝑡𝑑𝑒𝑝𝑎𝑟𝑡𝑢𝑟𝑒 ) = 𝑟⃗𝐸𝑎𝑟𝑡ℎ

(8)

𝑟⃗2 (𝑡2 ) = 𝑟⃗(𝑡𝑎𝑟𝑟𝑖𝑣𝑎𝑙 ) = 𝑟⃗𝑀𝑎𝑟𝑠

(9)

Figure 3 – Geometry of Lambert problem for an interplanetary trajectory. Source: Kemble, p. 9.9

11 The innovative conjecture Lambert had developed is based on geometric reasoning. In fact, Lambert believed that the time required to transverse an elliptic arc between two points depends only on the size of the orbit, i.e.semimajor axis 𝑎, on the chord length 𝑐 and the sum of the radii from the focus to points P1 and P2, or 𝑟1 + 𝑟2 . Mathematically, this can be expressed as follows: 𝑡2 − 𝑡1 = 𝑓(𝑎, 𝑐, 𝑟1 + 𝑟2 )

(10)

Starting from Kepler’s equation, Eq. (11), it is possible to derive an equation, sometimes called Lambert’s equation, Eq. (12), that does not depend on eccentricity and that is used to solve Lambert’s Problem. Such derivation can be found in most astrodynamics textbooks such as Orbital Mechanics by Prussing and Conway, pages 67-70. 10 𝜇 (𝑡 𝑎3 2

𝐸2 − 𝐸1 − 𝑒(sin 𝐸2 − sin 𝐸1 ) = √

− 𝑡1 )

(11)

𝑎3⁄2 [𝛼 − sin 𝛼 − (𝛽 − sin 𝛽)] = √𝜇(𝑡2 − 𝑡1 )

(12)

In Eq. (12), the unknown to solve for is semimajor axis 𝑎 while 𝛼 and 𝛽 are defined as: 𝛼 2

𝑠 1/2 2𝑎

(13)

𝛽 2

𝑠−𝑐 1/2 ) 2𝑎

(14)

𝑠 = (𝑟1 + 𝑟2 + 𝑐)

(15)

𝑐 = √𝑟1 2 + 𝑟2 2 − 𝑟1 𝑟2 cos Δ𝜃

(16)

sin ( ) = ( ) sin ( ) = ( 1 2

𝑟⃗ ̇𝑟⃗

Δ𝜃 = {

̂] ≥ 0 𝑖𝑓 [(𝑟⃗1 𝑥 𝑟⃗2 ) ̇𝐾

cos −1 ( 𝑟1 𝑟2 ) 1 2

𝑟⃗ ⋅𝑟⃗

̂] < 0 2𝜋 − cos−1 ( 𝑟1 𝑟2 ) 𝑖𝑓 [(𝑟⃗1 𝑥 𝑟⃗2 ) ̇𝐾 1 2

(17) Note that, given the definitions of 𝛼, 𝛽 and 𝑠, it can be seen that, as expected, Eq. (12) depends only on 𝑎, 𝑐, and 𝑟1 + 𝑟2 .

12 Derivation of Lambert’s Problem Solution Using the Universal Variable The solution to Lambert’s Problem gives the departure and arrival velocity vectors, 𝑣⃗1 and 𝑣⃗2 in Figure 3, that an arbitrary spacecraft needs to achieve in order to perform the transfer trajectory from P1 to P2 for a given amount of time. In this analysis, Lambert’s Problem was applied to the heliocentric cruise neglecting the sizes of the departure and arrival planets’ spheres of influence. Additionally, only prograde orbital transfers were considered. In order to obtain 𝑣⃗1 and 𝑣⃗2 in the J2000 ecliptic reference frame, Langrange coefficients can be used. In fact, starting from the general relationship involving Lagrange coefficients 𝑓, 𝑓̇, 𝑔, 𝑔̇ 𝑓 𝑟⃗ { 2} = [ ̇ 𝑣⃗2 𝑓

𝑔 𝑟⃗1 ]{ } 𝑔̇ 𝑣⃗1

(18)

the velocities 𝑣⃗1 and 𝑣⃗2 can be expressed as functions of the boundary conditions (i.e. the position vectors 𝑟⃗1 and 𝑟⃗2 ) as follows: 1

𝑣⃗1 = 𝑔 (𝑟⃗2 − 𝑓𝑟⃗1 )

(19)

1

𝑣⃗2 = 𝑔 (𝑔̇ 𝑟⃗2 − 𝑟⃗1 )

(20)

The Lagrange coefficients can be found using the following relationships. Note that 𝑣⃗1 and 𝑣⃗2 do not involve 𝑓̇. Hence, 𝑓̇ was not taken into account. 𝑓 =1−

𝑦(𝑧) 𝑟1

(21)

𝑦(𝑧) 𝜇

𝑔 = 𝐴√

𝑔̇ = 1 −

(22)

𝑦(𝑧) 𝑟2

(23) 𝑟 𝑟

1 2 𝐴 = ±sin Δ𝜃 √1−cos Δ𝜃

𝑦(𝑧) = 𝑟1 + 𝑟2 + 𝐴

𝑧𝐶3 (𝑧)−1 √𝐶2 (𝑧)

(24) (25)

13 The sign of 𝐴 in Eq. (24) is assessed in the next section, ‘Minimum Energy Solution.’ In Eq. (25), 𝐶3 (𝑧) and 𝐶2 (𝑧) are Stumpff functions and are defined by the following expressions:

𝐶2 (𝑧) =

𝐶3 (𝑧) =

1−cos √𝑧 𝑧 cosh √−𝑧−1 −𝑧 1 { 2 √𝑧−sin √𝑧 𝑧 3/2 sin √−𝑧−√−𝑧 (−𝑧)3/2 1 { 6

𝑖𝑓 𝑧 > 0 𝑖𝑓 𝑧 < 0

(26)

𝑖𝑓 𝑧 = 0 𝑖𝑓 𝑧 > 0 𝑖𝑓 𝑧 < 0

(27)

𝑖𝑓 𝑧 = 0

Moreover, 𝑧 is a variable defined as 𝑧 = 𝛼̃𝜒 2

(28)

1

𝛼̃ = 𝑎

(29)

In order to solve for the value of 𝑧, it is necessary to find the zeroes of the following transcendental function: 𝑦(𝑧) 3/2 ] 𝐶3 (𝑧) + 2 (𝑧)

𝐹(𝑧) = [𝐶

𝐴√𝑦(𝑧) − √𝜇(𝑡2 − 𝑡1 )

(30)

i.e. solving the equation 𝐹(𝑧) = 0, which can be done by using a root finding method such as the Newton-Raphson method. Once a guess on the initial value of 𝑧 is made, then the following equation can be solved iteratively until a certain relative tolerance is satisfied: 𝑧𝑖+1 = 𝑧𝑖 − 𝐹(𝑧𝑖 )/𝐹′(𝑧𝑖 ) Here, 𝐹′(𝑧𝑖 ) =

𝜕𝐹(𝑧𝑖 ) . 𝜕𝑧𝑖

(31)

Note that a good initial guess for 𝑧 would be 𝑧 = 0. Additionally, in Eq.

(28) 𝜒 is known as the universal variable. Using 𝜒 when solving Lambert’s problem is preferable because it avoids encountering discontinuities when iterating to solve for 𝑧. In fact, the solution to Lambert’s problem can yield an elliptical or hyperbolic transfer trajectory, depending on the given departure and arrival times. Because of this fact, while iterating to solve for 𝑧, the value of

14 semimajor axis 𝑎 could vary between positive, negative, and undefined. If “regular variables” instead of the universal variable were to be used, this latter possible value that 𝑎 could attain would cause the iterative process for finding 𝑧 to be inconclusive. Therefore, no solution for the Lambert’s Problem would be found. Once the value of 𝑧 is determined within a certain tolerance, then the velocity vectors 𝑣⃗1 and 𝑣⃗2 can be computed using the Lagrange coefficients as mentioned previously. The next sections present a special case of the minimal energy Lambert’s Problem solution and a summarized step-by-step algorithm for solving the Lambert’s Problem.

Minimum Energy Solution Additionally, in order to resolve the sign of Eq. (24), the minimum energy transfer must be assessed. In fact, the minimum transfer energy for the Lambert’s solution happens when the 𝜇

characteristic orbital energy 𝜀 = − 2𝑎 is minimum. It can be shown that 𝜀 is minimized when 𝑆

𝑎𝑚 = 2

(32)

Note that the subscript m stands for “minimum.” Substituting Eq. (32) into Eq. (13) and solving for 𝛼 yields to 𝛼𝑚 = 𝜋. Using this value for 𝛼𝑚 in Eq. (14) and solving for 𝛽 results in 𝛽𝑚 = 𝑠−𝑐 . 𝑠

2 sin−1 √

Using 𝛼𝑚 and 𝛽𝑚 in Eq. (12) permits to solve for (𝑡2 − 𝑡1 )𝑚 : 1

Δ𝑡𝑚 = (𝑡2 − 𝑡1 )𝑚 =

1 𝑠3 2 ( ) (𝜋 √𝜇 8

− 𝛽𝑚 + sin 𝛽𝑚 )

(33)

Note that minimizing the energy of the transfer orbit does not result in the minimum time of flight. Additionally, the values for semilatus rectum and eccentricity of the minimum energy transfer can be determined as follows: 𝑝𝑚 =

𝑟1 𝑟2 (1−cos Δ𝜃) 𝑐

(34)

15 𝑝

𝑒𝑚 = √1 − 𝑎𝑚

(35)

𝑚

Once the value of Δ𝑡𝑚 is determined, then for a given time of flight (𝑡2 − 𝑡1 ), 𝐴 from Eq. (24) can be determined as follows: 𝑟 𝑟

1 2 sin Δ𝜃 √1−cos Δ𝜃

𝐴=

𝑟 𝑟

{

1 2 −sin Δ𝜃 √1−cos Δ𝜃

𝑖𝑓 Δ𝑡𝑚 ≥ (𝑡2 − 𝑡1 ) (36) 𝑖𝑓 Δ𝑡𝑚 < (𝑡2 − 𝑡1 )

Algorithm for Solving Lambert’s Problem This section illustrates a step-by-step algorithm to solve the general case of the Lambert’s Problem for an interplanetary transfer from Earth to Mars. Prograde motion is assumed for all possible transfer orbits. Appendix A comprises of a flowchart that illustrates how this algorithm can be implemented in a computer language such as MATLAB in order to make computations such as the iterative process to find 𝑧 faster. Solution to Lambert’s Problems for given departure and arrival dates: 1. 𝑡𝑑𝑒𝑝 and 𝑡𝑎𝑟𝑟 are given 2. Compute 𝑟⃗𝐸𝑎𝑟𝑡ℎ , 𝑟⃗𝑀𝑎𝑟𝑠 , using the method described in Chapter 2. 3. Find the values of |𝑟⃗𝐸𝑎𝑟𝑡ℎ |, |𝑟⃗𝑀𝑎𝑟𝑠 | (lengths of the position vectors), chord 𝑐, transfer angle Δ𝜃, 𝑠 and 𝐴 4. Iterate to get the value of 𝑧 5. Solve for 𝑦(𝑧) 6. Find the Lagrange coefficients: 𝑓, 𝑔, 𝑔̇ 7. Compute 𝑣⃗1 and 𝑣⃗2

16 Planetary Departure Once 𝑣⃗1 and 𝑣⃗2 are determined via the solution to Lambert’s Problem, planetary departure and arrival must be considered. In fact, when setting up Lambert’s Problem, the spheres of influence of Earth and Mars were ignored. In reality, before being able to leave Earth’s sphere of influence, it is necessary to transit from a parking elliptical orbit about Earth to a hyperbolic orbit as shown in Figure 4.

Figure 4 – Departure orbit geometry. Source: Curtis, p. 443.7

Hence, the velocity of the spacecraft’s hyperbolic orbit at the boundary of the sphere of influence with respect to Earth must be: 𝑣⃗∞/𝐸𝑎𝑟𝑡ℎ = 𝑣⃗𝑠𝑐/𝑆𝑢𝑛,1 − 𝑣⃗𝐸𝑎𝑟𝑡ℎ/𝑆𝑢𝑛

(37)

17 Here, 𝑣⃗𝑠𝑐/𝑆𝑢𝑛,1 = 𝑣⃗1 which was calculated from the Lambert’s Problem while 𝑣⃗𝐸𝑎𝑟𝑡ℎ/𝑆𝑢𝑛 = 𝑣⃗𝐸𝑎𝑟𝑡ℎ which is the velocity of Earth with respect to the Sun. The latter quantity can be obtained from Standish’s algorithm as explained in Chapter 2. Therefore, it is possible to compute 𝑣⃗∞/𝐸𝑎𝑟𝑡ℎ and its magnitude. Additionally, note that Eq. (37) is a vector relation and therefore comparing the various velocity magnitudes does not suffice and is only applicable when all vectors are parallel to each other. In order to determine the necessary Δv to put the spacecraft onto a hyperbolic trajectory, initial orbital conditions of the spacecraft parking orbit must be given as shown on Figure 4. Assuming a circular parking orbit around the Earth with known altitude ℎ𝐸𝑎𝑟𝑡ℎ , the velocity of the spacecraft can be calculated as follows: 𝑣𝑐,𝐸𝑎𝑟𝑡ℎ = √𝑅

𝜇𝐸𝑎𝑟𝑡ℎ

(38)

𝐸𝑎𝑟𝑡ℎ +ℎ𝐸𝑎𝑟𝑡ℎ

Then, using the vis-viva equation, it is possible to write an energy balance between the parking orbit and the hyperbolic orbit: 2

(𝑣∞/𝐸𝑎𝑟𝑡ℎ ) 2

2

=

(𝑣𝑐,𝐸𝑎𝑟𝑡ℎ +∆𝑣1 ) 2

−𝑅

𝜇𝐸𝑎𝑟𝑡ℎ

(39)

𝐸𝑎𝑟𝑡ℎ +ℎ𝐸𝑎𝑟𝑡ℎ

In Eq. (39) the only unknown is ∆𝑣1 . Solving for ∆𝑣1 yields to two different solutions only one of 2

which is positive and is hence the correct answer. Usually, the term (𝑣∞/𝐸𝑎𝑟𝑡ℎ ) is referred to as 𝐶3. More generally, 2

𝐶3 = (𝑣∞/𝐷𝑒𝑝𝑎𝑟𝑡𝑢𝑟𝑒 𝑃𝑙𝑎𝑛𝑒𝑡 )

(40)

𝐶3 is therefore related to the characteristic energy of the departure hyperbolic orbit. Additionally, 𝐶3 is a key parameter used in interplanetary trajectory design as discussed in Chapter 4.

18 Planetary Arrival After the spacecraft travels from Earth to the sphere of influence of Mars, another maneuver must be performed in order to have the spacecraft be captured by the arrival planet. In fact, as Figure 5 shows, if no ∆𝑣2 is executed the spacecraft would do a flyby of Mars without entering a closed elliptical orbit around it and leaving its sphere of influence on a flyby trajectory.

Figure 5 – Planetary approach. Source: Curtis, p. 451.7

From the solution to Lambert’s Problem, it is possible to obtain 𝑣⃗∞/𝑀𝑎𝑟𝑠 as follows: 𝑣⃗∞/𝑀𝑎𝑟𝑠 = 𝑣⃗𝑠𝑐/𝑆𝑢𝑛,2 − 𝑣⃗𝑀𝑎𝑟𝑠/𝑆𝑢𝑛

(41)

19 Similarly to the previous section, 𝑣⃗𝑠𝑐/𝑆𝑢𝑛,2 = 𝑣⃗2 and 𝑣⃗𝑀𝑎𝑟𝑠/𝑆𝑢𝑛 can be computed with the method described in Chapter 2. Thus, after computing 𝑣⃗∞/𝑀𝑎𝑟𝑠 using Eq. (41), it is possible to solve for ∆𝑣2 from the following equation: 2

(𝑣∞/𝑀𝑎𝑟𝑠 ) 2

2

=

(𝑣𝑐,𝑀𝑎𝑟𝑠 +∆𝑣2 ) 2

−𝑅

𝜇𝑀𝑎𝑟𝑠

𝑀𝑎𝑟𝑠 +ℎ𝑀𝑎𝑟𝑠

(42)

Note that the above expression is true for a ∆𝑣2 performed at the hyperbolic periapse which corresponds to the desired final spacecraft parking orbit altitude, ℎ𝑀𝑎𝑟𝑠 . Additionally, 𝑣𝑐,𝑀𝑎𝑟𝑠 can be calculated as follows: 𝑣𝑐,𝑀𝑎𝑟𝑠 = √𝑅

𝜇𝑀𝑎𝑟𝑠

𝑀𝑎𝑟𝑠 +ℎ𝑀𝑎𝑟𝑠

(43)

In the approach section of the spacecraft journey from Earth to Mars, it should be noted that the main design parameter upon which the rest of a mission is constructed on is 𝑣∞/𝑀𝑎𝑟𝑠 .

Sphere of Influence Considerations As mentioned in Chapter 1, the spheres of influence of Earth and Mars are considered to be relatively small for the heliocentric transfer and relatively large for the planetary departure and arrival. The formula for computing the radius of the sphere of influence 𝑟𝑆𝑂𝐼 of a planet in the Solar System is given by: 𝜇𝑝𝑙𝑎𝑛𝑒𝑡 2/5

𝑟𝑆𝑂𝐼 ≈ 𝑟𝑝𝑙𝑎𝑛𝑒𝑡 (

𝜇𝑆𝑢𝑛

)

(44)

Here, 𝑟𝑝𝑙𝑎𝑛𝑒𝑡 is the semimajor axis of the planet’s orbit around the Sun. A derivation of Eq. (44) can be found in Wiesel on p. 320-322. When calculating 𝑟𝑆𝑂𝐼 for Earth and Mars, it can be found that: 𝑟𝑆𝑂𝐼,𝐸𝑎𝑟𝑡ℎ ≈ 924,652 𝑘𝑚 𝑜𝑟 145 𝑏𝑜𝑑𝑦 𝑟𝑎𝑑𝑖𝑖 𝑟𝑆𝑂𝐼,𝑀𝑎𝑟𝑠 ≈ 577,133 𝑘𝑚 𝑜𝑟 170 𝑏𝑜𝑑𝑦 𝑟𝑎𝑑𝑖𝑖

20 Hence, the assumptions made about the spheres of influence of Earth and Mars hold, especially if considering the energy balance expression for a hyperbolic to elliptic transfer, or vice-versa: 2

⃗⃗∞/𝑃𝑙𝑎𝑛𝑒𝑡 ) (𝑣 2



𝜇𝑃𝑙𝑎𝑛𝑒𝑡 𝑟𝑆𝑂𝐼

2

=

(𝑣𝑐,𝑃𝑙𝑎𝑛𝑒𝑡 +∆𝑣) 2

−𝑅

𝜇𝑃𝑙𝑎𝑛𝑒𝑡

𝑃𝑙𝑎𝑛𝑒𝑡 +ℎ𝑃𝑙𝑎𝑛𝑒𝑡

In Eq. (45) it can be noted that the term −

𝜇𝑃𝑙𝑎𝑛𝑒𝑡 𝑟𝑆𝑂𝐼

(45)

→ 0 as 𝑟𝑆𝑂𝐼 → ∞ or, for more practical

applications, as 𝑟𝑆𝑂𝐼 becomes large enough. Thus, Eq. (45) can be simplified to: 2

⃗⃗∞/𝑃𝑙𝑎𝑛𝑒𝑡 ) (𝑣 2

2

=

(𝑣𝑐,𝑃𝑙𝑎𝑛𝑒𝑡 +∆𝑣) 2

−𝑅

𝜇𝑃𝑙𝑎𝑛𝑒𝑡

𝑃𝑙𝑎𝑛𝑒𝑡 +ℎ𝑃𝑙𝑎𝑛𝑒𝑡

(46)

Total ΔV Calculation and Propellant Mass In order to obtain the total ∆𝑣 the overall Earth-Mars transfer requires, one can compute such quantity as follows: ∆𝑣𝑡𝑜𝑡 = ∆𝑣1 + ∆𝑣2

(47)

Note that ∆𝑣𝑡𝑜𝑡 does not usually represent the total ∆𝑣 budget for a mission. In fact, other minor maneuvers such as trajectory correction maneuvers were not taken into account in this analysis. Another key mission parameter is represented by propellant mass. A quick estimate of how much propellant is needed for given mission and propulsion system is given by: 𝑚 𝑝 = 𝑚 1 (1 − 𝑒 −∆𝑣𝑡𝑜𝑡 /𝑣𝑒𝑥 )

(48)

Here, 𝑚 1 represents the initial mass of the spacecraft (payload, structure, and propellant) while 𝑣𝑒𝑥 is the exhaust velocity of the propulsion system, or: 𝑣𝑒𝑥 = 𝐼 𝑠𝑝 ∗ 𝑔𝑆𝐿

(49)

where 𝑔𝑆𝐿 is the acceleration due to gravity at sea level on Earth. Without knowing the total mass of the spacecraft, it is still possible to rearrange Eq. (48) and solve for propellant mass as a percentage of the total mass:

21 𝑚𝑝

(

)

𝑚 1 𝑝𝑒𝑟𝑐𝑒𝑛𝑡𝑎𝑔𝑒

= (1 − 𝑒

𝑚𝑝

Examples of ∆𝑣 and ( Chapter 4.

)

∆𝑣 − 𝑡𝑜𝑡 𝑣𝑒𝑥

𝑚 1 𝑝𝑒𝑟𝑐𝑒𝑛𝑡𝑎𝑔𝑒

) ∗ 100%

for a give set of departure and arrival dates are given in

(50)

22

Chapter 4 Porkchop Plots Porkchop plots are computer-generated contour plots that display launch dates vs. arrival dates for characteristics of an interplanetary flight path for a given launch and arrival opportunity between two planets within the Solar System6. Perhaps the most important characteristics of porkchop plots are 𝐶3 at departure (or 𝐶𝐿3) and 𝑣∞ at arrival. In fact, by looking at a porkchop plot it is possible to choose the launch and arrival windows given specific spacecraft performance such as ∆𝑣 and 𝐼 𝑠𝑝 . Figure 6 shows an example of a porkchop plot where contour lines of constant 𝐶3 and arrival 𝑣∞ are displayed in red and blue respectively. Additionally, lines of constant time of flight are shown in green. Porkchop plots can be obtained by solving Lambert’s Problem for all the possible combination of departure vs. arrival dates provided that no arrival date is earlier than any departure date. Because such task is computationally time consuming, using a computer program such as MATLAB is essential. A flowchart in Appendix B shows how the solution to the Lambert’s Problem algorithm presented in Chapter 3 can be expanded in order to create porkchop plots. Considering the porkchop plot of Figure 6, one can pick departure and arrival dates, say July 30, 2020 and February 5, 2021 respectively, to determine some of the key parameters of the resulting orbital transfer. The two dates (the two dotted lines in Figure 6) define a specific point on the graph which, for convenience, was denoted by an orange dot. From the graph, it is visible that the orange dot lies close to the 14-level red line and the 2.7-level blue line. According to the plot legend, this means that for this particular combination of dates, 𝐶𝐿3 ≈ 14 𝑘𝑚2 /𝑠 2 and 𝑣∞ ≈ 2.7 𝑘𝑚/𝑠. Additionally, it is possible to estimate the time of flight to be about 190 days since the

23 dot lies slightly below the 200-level green line. In other words, for this 190 day transfer a characteristic energy of about 14 𝑘𝑚2 /𝑠 2 must be reached by the spacecraft as it is leaving Earth’s sphere of influence; also, upon arrival to Mars, the spacecraft would enter Mars’ sphere of influence with a speed of approximately 2.7 𝑘𝑚/𝑠. Note that in order to know more accurate results for 𝐶𝐿3, 𝑣∞ and time of flight, it is necessary to consult the numerical results that MATLAB gives.

Figure 6 – Porkchop plot for the years 2020-2022 timeframe. The orange dot represents the sample point used in this section to explain how to read a porkchop plot.

24 Key Features of a Porkchop Plot One of the key features of porkchop plots is the presence of a singularity shown on the plot as a diagonal line usually going from the bottom left corner to the upper right corner where the values of 𝐶3 and arrival 𝑣∞ get extremely large. This singularity is due to the fact that 0 and, most importantly, 180 degree transfers (i.e. Δ𝜃 = 0° and Δ𝜃 = 180°) make the transfer plane between departure and arrival undefined as explained in Chapter 6. Because of this fact, solutions of Lambert’s Problem close to the singularity give unpractical results for interplanetary trajectory design. Additionally, this singularity divides the possible transfer orbits into two main classes: type I transfers and type II transfers. The former are defined for having 0° < Δ𝜃 < 180° and are found in the region bounded above by the singularity diagonal; the latter are defined for having 180° < Δ𝜃 < 360° and are found in the region bounded below by the singularity diagonal. Examples of type I and type II transfers are shown in Figure 7 and Figure 8 respectively. Note that in these figures blue, red and green represent Earth’s orbit, Mars’ orbit, and the transfer orbit Earth-Mars respectively.

25

Figure 7 – Example of a Type I transfer with launch on September 8, 2020 and arrival on December 17, 2020.

26

Figure 8 – Example of a Type II transfer with launch on June 1, 2020 and arrival on May 16, 2021.

ΔV vs. Departure Date vs. Arrival Date Once 𝐶3 at departure and 𝑣∞ at arrival have been calculated for all the possible combinations of dates of interest, as explained in Chapter 3, it is possible to calculate ∆𝑣1 and ∆𝑣2 using Eq. (39) and Eq. (42). Then, using Eq. (47), ∆𝑣𝑡𝑜𝑡 can be found for all the range of dates. Figure 9 shows a contour plot of ∆𝑣𝑡𝑜𝑡 for the same ranges of dates used for the porkchop plot of Figure 6. In this case, an initial Earth-centered circular parking orbit with an altitude of 300 km and a final Mars-centered circular parking orbit with an altitude of 200 km were considered.

27

Figure 9 – ΔVtot for the 2020-2022 Earth-Mars transfer trajectories.

From the ∆𝑣𝑡𝑜𝑡 plot the transfer dates that require the least ∆𝑣 and hence the least amount of propellant can be found. For example, for the 2020-2022 timeframe Figure 9, Table 3 summarizes the key results. Notice that this particular value of ∆𝑣𝑡𝑜𝑡 is a minimum for the type I transfer orbit presented in Figure 9.

28 Table 3 – Key results of the Earth-Mars trajectory that minimizes ΔVtot in the 2020-2022 timeframe.

Earth-Mars Trajectories for the 2020-2022 timeframe – Minimum ΔV Parameter

Value 5.8921 𝑘𝑚/𝑠

∆𝒗𝒕𝒐𝒕 C3 at Launch

14.045 𝑘𝑚2 /𝑠 2

𝒗∞ at Arrival

2.5748 𝑘𝑚/𝑠

Departure Date

July 27, 2020

Arrival Date

February 19, 2021

Time of Flight

207 days

Additionally, it is worth noting that the combination of dates at which the minimum ∆𝑣𝑡𝑜𝑡 happens is usually not where the minimum C3 at launch and minimum 𝑣∞ at arrival occur, although lower values of C3 and 𝑣∞ tend to give lower values of ∆𝑣𝑡𝑜𝑡 .

Propellant Mass Knowing the required ∆𝑣𝑡𝑜𝑡 for the interplanetary transfer, propellant mass as a percentage of the total spacecraft mass can be easily obtained using Eq. (50). Additionally, as mentioned in Chapter 3, the 𝐼𝑠𝑝 of the propulsion system used for the main mission transfer must also be known (or estimated). Using an 𝐼𝑠𝑝 of 350 seconds and considering the 2020-2022 timeframe results into Figure 10. Notice that, as expected, the minimum propellant mass required occurs where the minimum ∆𝑣𝑡𝑜𝑡 occurs.

29

Figure 10 – Propellant mass as a percentage of total initial mass for the 2020-2022 timeframe.

30

Chapter 5 Orbital Parameters of the Transfer Trajectories and Examples In order to obtain additional information on the transfer trajectories, such as maximum and minimum distances from the Sun and inclination relative to the ecliptic plane, the orbital parameters of the transfer trajectories must be calculated. Recall that the five classical orbital elements are semimajor axis 𝑎, eccentricity 𝑒, inclination 𝑖, right ascension of the ascending node Ω, and argument of periapse 𝜔. Also, for the two-body problem, from the position and velocity vectors at a specific times it is possible to compute all of the five orbital elements for that particular trajectory. Amongst the orbital elements, those of most interest for this analysis are 𝑎, 𝑒, and 𝑖.

Eccentricity Eccentricity represents a key parameter especially when considering both elliptical (0 < 𝑒 < 1) and hyperbolic (𝑒 > 1) transfer trajectories. For given Earth-Mars geometry, spacecraft sent on hyperbolic trajectories from Earth to Mars would arrival at Mars faster than elliptical orbits. An unmanned spacecraft may not be required to reach Mars within a specific time constraint, elliptical orbits are preferred. On the other hand, a manned mission would be more likely to travel on much faster trajectory, particularly to avoid astronauts from being exposed to interplanetary radiation for long periods of time. Therefore, trajectories with 𝑒 > 1 should also be

31 considered. Figure 11 shows examples of elliptical-only trajectories while

Figure 12 gives examples where 𝑒 > 1.

32

Figure 11 – Example of Earth-Mars trajectories with e < 1.

33

Figure 12 – Example of Earth-Mars trajectories including e > 1.

Semimajor Axis In order to determine the size of the transfer orbit, semimajor axis 𝑎 must be known. Additionally, 𝑎 can be used in combination with 𝑒 to calculate the maximum and minimum distance from the Sun, i.e. apoapsis and periapsis distances respectively, as follows: 𝑟𝑎𝑝𝑜𝑎𝑝𝑠𝑒 = 𝑎(1 + 𝑒)

(51)

34 𝑟𝑝𝑒𝑟𝑖𝑎𝑝𝑠𝑒 = 𝑎(1 − 𝑒)

(52)

If, for example, a spacecraft relies primarily on solar power, it is necessary to know what the furthest distance from the Sun will be in order to be able to correctly size its solar panels. On the other hand, it is also important to know the minimum distance because if, for example, a mission where to use a particular transfer that gets much closer to the Sun than Earth is, the spacecraft involved in such mission may require additional radiation shielding. Figure 13 shows an example of various semimajor axis for a given set of dates.

Figure 13 – Example of various semimajor axes for Earth-Mars trajectories

35 Inclination Because Earth’s and Mars’ orbits are relative inclination is small, transfer orbits from Earth to Mars are also expected to have a small inclination as Figure 14 suggests for most of the departure-arrival combinations. Notice that as the date combinations approach the singularity discussed in the porkchop plot section, the inclinations of the transfer orbits increase reaching values in the range of 60° − 80°. This is due to the fact that as Δ𝜃 → 0° and as Δ𝜃 → 180° the orbital plane becomes undefined. In fact, the Sun, Earth, and Mars can be treated as three points in a three-dimensional space that define a plane, namely the orbital plane of the transfer orbit. When the two planets and the Sun are close to being lined up, the transfer planes take on inclination values that would not be practical for an actual mission and that would require very large amounts of propellant. In the realm of numerical solutions, it is nearly impossible to get a combination of departure-arrival dates that would give exactly Δ𝜃 = 0° or Δ𝜃 = 180°. On the other hand, values of Δ𝜃 that are close enough to 0° or 180° can make the variable 𝐴, which is defined by Eq. (24), become extremely large which yields to the large values of 𝐶3 at launch and arrival 𝑣∞ as previously discussed. Figure 15 and Figure 16 show cases of orbital transfers with relatively high inclination. In particular, Figure 16 is a case of a nearly 90° inclination. Such transfers are not reasonable because they require spacecraft to reach extremely high out-of-plane 𝑣∞ with respect to Earth (hence high values of 𝐶3) and they also do not fully take advantage of Earth’s orbital velocity. Therefore, getting a high inclination transfer as a result from Lambert’s problem is a signal that, as discussed later, the solution to Lambert’s problem fails to give a reasonable answer. Lastly, as discussed in Chapter 6, reasonable solutions for Δ𝜃 = 180° exist and are referred to as Hohmann transfers.

36

Figure 14 – Inclination of the transfer orbital plane for various Earth-Mars trajectories

37

Figure 15 – An example of an Earth-Mars trajectory with relatively high inclination.

38

Figure 16 – Example of an extremely high inclination orbital transfer from Earth to Mars.

39

Trajectory Examples This section presents the graphical representation of transfer trajectories Earth-Mars of various spacecraft that were launched in the past and that are planned to be launched in the near future. Table 4 summarizes what is presented in Figures 17 – 20. Table 4 – Summary of trajectories presented in this section.

Earth-Mars Trajectories of Various Spacecraft Spacecraft Name Mariner 41 Mars Reconnaissance Orbiter (MRO) 2 Mars Science Laboratory3

Launch Date

Arrival Date

Figure #

November 28, 1964

July 14, 1965

17

August 12, 2005

March 10, 2006

18

November 26, 2011

August 6, 2012

19

November 18, 2013*

September 22, 2014*+

20

Mars Atmosphere and Volatile EvolutioN (MAVEN) 5 *nominal dates; +expected date of Mars orbital insertion As Figures 17 – 20 show, spacecraft that have been launched to Mars follow trajectories that have a transfer angle Δ𝜃 close to 180°. In fact, as discussed in more details in Chapter 6, analytical solutions for the case Δ𝜃 = 180° exist and are known as Hohmann transfers.

40

Figure 17 – Mariner 4 trajectory.

41

Figure 18 – MRO trajectory.

42

Figure 19 – Mars Science Laboratory trajectory.

43

Figure 20 – MAVEN trajectory.

44 Extreme Cases Sometimes it is useful to consider cases of hyperbolic transfers. In fact, as previously mentioned in the Eccentricity section, hyperbolic transfers usually provide much shorter times of flight than elliptical orbits. As opposed to elliptical cases, when 𝑒 > 1 both 𝐶3 and 𝑣∞ increase drastically as time of flight decreases. Therefore, in order to use such transfers it is necessary to have a propulsion system capable of performing extremely high ∆𝑣 maneuvers in a relatively short amount of time, i.e. high thrust. Previous research, such as the NERVA project, has shown that nuclear propulsion systems would be capable of with specific impulse in the order of 9001000 seconds, which is at least twice as much as typical chemical rockets.11 Such a high 𝐼𝑠𝑝 would decrease the amount of propellant needed to accomplish the mission and allow for a bigger payload mass. Examples of low time of flight hyperbolic trajectories are shown in Figure 21 –24 and summarized in Table 5. Table 5 – Summary of some examples of hyperbolic trajectories with low transfer times.

Examples of Hyperbolic Earth-Mars Trajectories Launch Date

Arrival Date

TOF (days)

Eccentricity

Figure #

January 1, 2029

March 1, 2029

59

2.033

21

February 1, 2031

April 1, 2031

59

1.986

22

May 1, 2033

June 15, 2033

45

1.308

23

July 1, 2035

August 1, 2053

31

2.639

24

Note that the most propellant-convenient hyperbolic trajectories with low time of flight take place close to the dates of the opposition of Earth and Mars. In fact, when two planets are in opposition their distance is minimized. On the other hand, it would be extremely inconvenient to launch a spacecraft from Earth away from the Sun towards Mars since the spacecraft propulsion system would have to directly “fight” against the Sun’s gravitational force. Therefore, for the examples listed above, departure and arrival dates that are 30-45 days before and/or after the

45 opposition of the two planets were considered. For more details regarding the trajectories taken into account see Appendix C. Additionally, other hyperbolic trajectory cases are presented in Chapter 6.

46

Figure 21 – Hyperbolic Earth-Mars trajectory case 1.

47

Figure 22 – Hyperbolic Earth-Mars trajectory case 2.

48

Figure 23 – Hyperbolic Earth-Mars trajectory case 3.

49

Figure 24 – Hyperbolic Earth-Mars trajectory case 4.

50

Chapter 6 A Particular Case and Applications As previously discussed, the solution to Lambert’s Problem does not always give reasonable results in terms of values of 𝐶3, 𝑣∞ and inclination of the transfer orbit for cases close to Δ𝜃 = 0° or Δ𝜃 = 180°. Making the rational assumption that Earth and Mars have coplanar circular orbits, it is possible to analyze the Δ𝜃 = 180° case analytically. This particular case is the classic Hohmann transfer.

The Hohmann Transfer The case of the Hohmann transfer is particular since the transfer angle between the departure planet and arrival planet is considered to be exactly 180°. Additionally, it is well known that the Hohmann transfer is the most energy efficient two-burn maneuver for transferring between two coplanar circular orbits sharing a common focus. This makes the Hohmann transfer the most propellant efficient transfer between two planets orbiting the Sun. It also must be noted that analytical solutions to find the total ∆𝑣 for Hohmann transfers exist with the additional assumption that Earth and Mars have circular and coplanar orbits. Knowing the initial (Earth) and final (Mars) orbital parameters of a desired spacecraft and the propulsion system 𝐼𝑠𝑝 , an algorithm that solves the Hohmann transfer problem can be implemented (in this case MATLAB was used). An example of input and output is summarized in Table 6. For launch opportunities and arrival dates for future Hohmann transfers refer to Table 7.

51 Table 6 – Example of a Hohmann transfer from Earth to Mars

Hohmann Transfer Example Input Parameters Initial Earth orbit altitude

Final Mars orbit altitude

Specific Impulse

300 km

5982 km*

350 seconds

Results Departure ΔV

Arrival ΔV

TOF

Propellant mass %

3.590 km/s

1.881 km/s

259 days

79.68%

*this altitude corresponds to the average distance of the Martian moon Phobos from Mars. Although the Hohmann transfer is the most fuel efficient orbital maneuver between two circular and coplanar orbits (the total ∆𝑣 is about 5.5 – 5.6 𝑘𝑚/𝑠), it is not considered to be a “fast transfer.” In fact, while a Hohmann transfer from Earth to Mars requires a time of flight of approximately 260 days, it is possible to create faster trajectories to accomplish the same mission in much less time as discussed in Chapter 5 in the section Extreme Cases. Faster trajectories involving higher ∆𝑣 maneuvers are discussed next.

Application of Lambert’s Problem to Fast Trajectories As previously discussed, given a propulsion system with high enough specific impulse, it would possible to utilize faster trajectories in a way that would make interplanetary manned missions to Mars possible. In fact, two of the primary limiting factors in a manned mission to Mars are mission duration and amount of propellant available to accomplish such mission. The former is a direct consequence of the biological response of the human body to interplanetary radiation and microgravity as well as the limited amount of resources, such as food and oxygen, that can be transported to sustain the crew. The latter mainly depends on the propulsion system used and the ∆𝑣 maneuvers that need to be performed in order to leave Earth’s sphere of

52 influence and to establish a closed orbit around Mars. While the development of more efficient propulsion systems is most definitely a necessity for manned missions to Mars, selecting a trajectory that would minimize both time of flight and total ∆𝑣 (and hence minimize propellant mass) simultaneously is also a key decision in designing the mission. For the following analysis, only the outbound Earth-Mars trajectory was considered. A similar analysis can be done for the return trip Mars-Earth. Furthermore, only trajectories with a time of flight lower than 150 days (𝑇𝑂𝐹𝑚𝑎𝑥 = 150) were taken into account. Additionally, only reasonable orbital transfers were considered by setting an upper bound to maximum ∆𝑣𝑡𝑜𝑡 as a function of 𝐼𝑠𝑝 ∆𝑣𝑡𝑜𝑡,𝑚𝑎𝑥 = 𝑔0 𝐼𝑠𝑝 ln ( 𝑚𝑝

where 𝑅𝑝 = (𝑚 ) 1

𝑚𝑎𝑥

1 ) 1−𝑅𝑝

(53)

, i.e. the maximum propellant to total mass ratio to be considered for the

mission. For this analysis, 𝑅𝑝 was varied between 0.85 and 0.9 and 𝐼𝑠𝑝 was kept constant at a value of 1000 seconds. Note that examples of fast transfer were presented in the previous section, although they are not yet technologically feasible due to their extremely high ∆𝑣𝑡𝑜𝑡 (refer to Appendix C for additional details). Theoretically, there are an infinite amount of possible trajectories from Earth to Mars, but only those occurring up to about two months before and two months after an Earth-Mars opposition were considered. This is due to the fact that this situation results in the shortest distances Earth-Mars. Oppositions occurring in the future are listed in Table 7.

53 Table 7 – Earth-Mars oppositions and launch opportunities, 2014-2036. Launch opportunities and arrival dates assume a Hohmann transfer.[15]

Launch Opportunity

Opposition

Arrival Date

12/31/13

4/7/14

9/16/14

2/15/16

5/22/16

10/31/16

4/20/18

7/26/18

1/4/19

7/8/20

10/13/20

3/24/21

9/1/22

12/7/22

5/18/23

10/10/24

1/15/25

6/26/25

11/14/26

2/19/27

7/31/27

12/18/28

3/25/29

9/3/29

1/25/31

5/2/31

10/11/31

3/22/33

6/27/33

12/6/33

6/10/35

9/15/35

4/24/36

In order to identify the most effective trajectories in terms of time of flight (𝑇𝑂𝐹, in seconds) and ∆𝑣𝑡𝑜𝑡 , the following efficiency parameter is introduced 𝜂=

1 𝑇𝑂𝐹∗∆𝑣𝑡𝑜𝑡

(54)

Therefore, it is now possible to compare various orbital transfers using 𝜂. In fact, trajectories with higher values of 𝜂 are preferable. Next are some examples transfers with TOF less than 150 days. Figure 25 shows an example of a contour plot of trajectory efficiency 𝜂 (in blue) around the 2029 Earth-Mars opposition. In red and green are ∆𝑣𝑡𝑜𝑡,𝑚𝑎𝑥 and 𝑇𝑂𝐹𝑚𝑎𝑥 respectively; all possible admissible solutions are therefore constrained to be under the green line and above the red line (in Figure 25, only solutions to the right of the red and green line crossing are possible).

54

Figure 25 – Contour plot of trajectory efficiency η centered about the 2029 opposition.

Note that usually ∆𝑣𝑡𝑜𝑡 goes up as 𝑇𝑂𝐹 goes down, hence making the maximization of 𝜂 not trivial. Additionally, the regions on the graph where 𝜂 is not smooth is due to the changing from 𝑒 < 1 to 𝑒 > 1. In fact, the solution to Lambert’s problem close to 𝑒 = 1 requires more iterations to converge and is sometimes slightly inaccurate from a numerical perspective. Two additional examples of trajectory efficiency plots are presented in Figures 26 and Figure 27.

55

Figure 26 – Contour plot of trajectory efficiency η centered about the 2033 opposition.

56

Figure 27 – Contour plot of trajectory efficiency η centered about the 2035 opposition.

Details about the solutions with the highest 𝜂 for the three examples presented are summarized on Table 8. Furthermore, the trajectories of such optimized trajectories are given in Figures 28-30.

57 Table 8 – Earth-Mars fast transfers with highest efficiency for 𝑻𝑶𝑭𝒎𝒂𝒙 = 𝟏𝟓𝟎 days and 𝑰𝒔𝒑 = 𝟏𝟎𝟎𝟎 seconds.

Examples of Reasonable Earth-Mars Fast Transfers with η = ηmax Launch Date

Arrival Date

TOF (days)

∆𝒗𝒕𝒐𝒕 (𝒌𝒎/𝒔)

Figure #

December 31, 2028

May 20, 2029

150

11.051

28

May 1, 2033

September 15, 2033

137

7.410

29

August 3, 2035

November 10, 2035

99

9.000

30

58

Figure 28 – Earth-Mars optimized fast transfer for the year 2028-2029.

59

Figure 29 – Earth-Mars optimized fast transfer for the year 2033.

60

Figure 30 – Earth-Mars optimized fast transfer for the year 2035.

61

Conclusion and Future Work

The analyses and results presented in this thesis give an idea of the difficulties a mission from Earth to Mars can present from an astrodynamics point of view. Some of these challenges include the ∆𝑣 maneuvers (and hence the amount of propellant) required to leave Earth’s sphere of influence and to establish a closed orbit around Mars upon arrival and the time of flight needed for the transfer orbit to take place. This last one is especially critical when discussing human missions. In this thesis, some important and useful tools to analyze and decide mission parameters related to the outbound Earth-Mars possible trajectories are discussed. One of these, and perhaps the most important mentioned throughout the thesis, is porkchop plots. Knowing the position and velocity vectors of the departure and arrival planets for given departure and arrival dates, it is possible to solve Lambert’s problem. Many methods for solving Lambert’s problem exist and the one presented in Chapter 3 makes use of the universal variable. From the solution to Lambert’s problem and knowing the desired initial and final parking orbits about Earth and Mars respectively, porkchop plots can be created. Space agencies such as NASA and ESA create and utilize porkchop plots to determine the possible launch and arrival windows for various missions, including missions to Mars. In fact, given the launch vehicle and upper stage capabilities, the scientific mission requirements and the results of porkchop plots, nominal launch and arrival dates can be established. Along with the optimal type I and type II orbital transfers (i.e. those transfers whose ∆𝑣 is minimized) results from Lambert’s problem can help determine the geometry of the transfer orbit. As shown in various examples of trajectories presented in Chapter 5 and Chapter 6, orbital elements such as eccentricity, inclination and semimajor axis can vary greatly. In fact, for

62 example, since Lambert’s solution fails close to orbital transfers of 180°, trajectories with extremely high inclinations should not be considered. This can be seen from the numerical results obtained using software such as MATLAB or via visual results of the actual trajectory. Given the current state of the art chemical propulsion systems, a manned mission to Mars would not be feasible. As discussed in Chapter 5 and Chapter 6, assuming that propulsion systems with high specific impulse, such as nuclear thermal rockets, could be used, manned missions to Mars would become possible. Based on the efficiency parameter defined in Chapter 6, trajectories which are considered optimal in terms of both transfer time and total ∆𝑣 can be found. For example, for the 2035 case, the optimal trajectory has a ∆𝑣𝑡𝑜𝑡 of about 9.0 𝑘𝑚/𝑠, which is approximately 3.5 𝑘𝑚/𝑠 more ∆𝑣 than a Hohmann transfer, although it requires a time of flight of only 99 days as opposed to the 260 days required by the Hohmann transfer. In fact, this particular trajectory has a much higher ∆𝑣/𝑑𝑎𝑦 ratio than a classic Hohmann transfer. Thus, while increasing the value of ∆𝑣𝑡𝑜𝑡 (still to a reasonable amount), the time of flight to Mars is drastically lowered. It must be noted the porkchop plots and related results obtained throughout this thesis would be part of a preliminary mission design. In order to get more details about an actual trajectory for a mission to Mars, more factors, such as the type of spacecraft to consider, would have to be included. For example, landing a rover, such as Curiosity, onto the Martian surface would be a much different mission than getting an orbiter, such MRO, into a science orbit about Mars. Due to the presence of an atmosphere and the complicated EDL procedures required to arrive safely onto the Martian surface, the first manned missions towards Mars might result in the exploration of the Martian moons, Phobos and Deimos. Future work would then include B-plane targeting in order to intercept the moons of Mars and how to launch from Earth such that the arrival orbital plane in the Martian sphere of influence would be close to that of one of the moons.

63

Appendix A Lambert Solution – Flowchart The following flowchart outlines the key steps in solving the Lambert Problem.

Figure A 1 – Flowchart showing the main steps when solving Lambert’s problem.

64

Appendix B Creating Porkchop Plots – Flowchart

65

Figure B 1 – Flowchart showing the main steps when creating porkchop plots.

66 Appendix C Trajectory Details The following table presents the values of the main parameters of the various trajectories of missions conducted in the past.

Table C 1 – Earth-Mars trajectories of spacecraft sent to Mars in the past and MAVEN. All of the parameters were computed using MATLAB scripts derived from those used to compute porkchop plots.

Earth-Mars Trajectories of Various Spacecraft Calculated

Mariner 4

MRO

MSL

MAVEN

Parameters/

11/28/1964 –

8/12/2005 –

11/26/2011 –

11/18/2013 –

Spacecraft

7/14/1965

3/10/2006

8/6/2012

9/18/2014

Launch C3 (𝒌𝒎𝟐 /𝒔𝟐 )

9.85

16.33

10.68

12.08

4.4502

2.8384

3.5423

3.1723

228

210

254

304

Semimajor axis (𝑨𝑼)

1.2777

1.3473

1.2607

1.2074

Eccentricity

0.228

0.248

0.219

0.199

0.2061

3.0125

1.4952

2.0191

356.68

358.41

170.67

29.45

65.14

-40.68

243.15

55.66

1.5548

1.6102

1.5368

1.4474

Arrival 𝒗∞ (𝒌𝒎/𝒔) Time of flight (𝒅𝒂𝒚𝒔)

Inclination (𝒅𝒆𝒈𝒓𝒆𝒆𝒔) Argument of periapse (𝒅𝒆𝒈𝒓𝒆𝒆𝒔) RAAN (𝒅𝒆𝒈𝒓𝒆𝒆𝒔) Maximum distance from the Sun

67

The following table presents the values of the main parameters of the various trajectories of hypothetical hyperbolic transfers for potential future missions.

Table C 2 – Earth-Mars trajectories of hypothetical future missions. All of the parameters were computed using MATLAB scripts derived from those used to compute porkchop plots.

Earth-Mars Trajectories of Hypothetical Future Missions Calculated

Case 1

Case 2

Case 3

Case 4

Parameters/

(1/1/2029 –

(2/1/2031 –

(5/1/2033 –

(7/1/2035 –

Spacecraft

3/1/2029)

4/1/2031)

6/15/2033)

8/1/2035)

Launch C3 (𝒌𝒎𝟐 /𝒔𝟐 )

496.21

495.77

273.64

748.53

Arrival 𝒗∞ (𝒌𝒎/𝒔)

34.2260

34.5374

26.1093

35.6000

Time of flight (𝒅𝒂𝒚𝒔)

59

59

45

31

Semimajor axis (𝑨𝑼)

-0.9478

-0.9894

-3.1931

-0.6176

2.033

1.986

1.308

2.639

1.5952

0.7205

1.5695

2.6220

6.09

9.78

163.34

173.66

100.58

131.72

40.59

98.65

1.6420

1.5998

1.4561

1.3831

Eccentricity Inclination (𝒅𝒆𝒈𝒓𝒆𝒆𝒔) Argument of periapse (𝒅𝒆𝒈𝒓𝒆𝒆𝒔) RAAN (𝒅𝒆𝒈𝒓𝒆𝒆𝒔) Maximum distance from the Sun

In order to produce the results shown in this appendix, various MATLAB scripts based on the flowchart presented in Appendix B were used.

68

BIBLIOGRAPHY 1. "Mariner 3 & 4." Mars Exploration Program. NASA JPL, n.d. Web. 10 Oct. 2013. . 2. "Mars Reconnaissance Orbiter." Mars Reconnaissance Orbiter. NASA JPL, n.d. Web. 10 Oct. 2013. . 3. "Mars Science Laboratory." Mars Science Laboratory. NASA JPL, n.d. Web. 10 Oct. 2013. . 4. "MATLAB." MATLAB - The Language of Technical Computing. MathWorks, n.d. Web. 10 Sept. 2013. . 5. "MAVEN." Mars Exploration Program. NASA JPL, n.d. Web. 10 Oct. 2013. . 6. "NASA's Mars Exploration Program." Mars Exploration: Features. NASA JPL, n.d. Web. 10 Nov. 2013. . 7. Curtis, Howard D. Orbital Mechanics for Engineering Students, 2nd edition. Elsevier Aerospace Engineering Series, 2010. 8. Davis, Phil. "Solar System Exploration: Missions: In Flight." Solar System Exploration: Missions: In Flight. NASA, n.d. Web. 7 Aug. 2013. . 9. Kemble, Stephen. Interplanetary Mission Analysis and Design. Berlin: Springer in Association with Praxis, 2006. Print. 10. Prussing, John E., and Bruce A. Conway. Orbital Mechanics. New York: Oxford UP, 1993. Print.

69 11. Robbins, W.H. and Finger, H.B., "An Historical Perspective of the NERVA Nuclear Rocket Engine Technology Program", NASA Contractor Report 187154/AIAA-91-3451, NASA Lewis Research Center, NASA, July 1991. 12. Standish, Erland M. Keplerian Elements for Approximate Position of the Major Planets. N.p.: NASA JPL, 2006. Print. 13. Vallado, David A. Fundamentals of Astrodynamics and Applications. 3rd ed. Dordrecht: Kluwer Academic, 2007. Print. 14. Yeomans, Donald K. "Keplerian Elements for Approximate Positions of the Major Planets."Keplerian Elements for Approximate Positions of the Major Planets. NASA JPL, n.d. Web. 09 Aug. 2013. . 15. Wertz, James R., David F. Everett, and Jeffery Puschell. Space Mission Engineering: The New SMAD. Page 207 Microcosm Press and Springer, 2011.

ACADEMIC VITA Davide Conte [email protected] Via Laviosa 12/19 Genova Pegli, 16156 Italy ________________________________________ Education The Pennsylvania State University, University Park, PA 16802

2009 – 2014

Schreyer Honors College, Integrated Undergraduate/Graduate (IUG) Candidate 

M.S. in Aerospace Engineering

2013 – 2014



B.S. in Aerospace Engineering

2009 – 2014



B.S. in Mathematics

2009 – 2014

Honors and Awards o

Participated in RASC-AL 2013 (sponsored by NASA, NIA, and Boeing) as team leader of Martian Lion 

Presented an innovative mission design for future human exploration of Mars through a written report, a presentation, and a poster session; conducted outreach events in local middle schools and Boy Scout troops to motivate young students to get involved in STEM fields with a focus on space exploration

o

Space Systems Engineering Certificate (SPSYS)

o

First place winner, College of Engineering Research Symposium poster competition

o

Dean’s List recognition every semester since Fall 2009

o

Ju-jitsu black belt (Shodan), certified experienced instructor and referee

o

Annually take part into the Dante Prize sponsored by the Dante Society of America

o

Scholarships received in Italy: 3 from private institutions and 9 from Government (tot. of about $ 14,000)

o

GeoEye Foundation Scholarship for the academic year 2012 – 2013

o

Donald G. and Jayne L. Steva Scholarship for the academic year 2013 – 2014

Association Memberships/Activities o

Sigma Gamma Tau – Penn State Aerospace Honors Society member

o

Penn State Astronomy Club – Secretary (2010 – 2013): manage club affairs and take notes at meetings

o

Group Leader and Mentor for Schreyer Honors College Orientation 2011 and 2012: 

Lead groups of incoming first-year students through various orientation activities

Professional Experience o

Teaching Intern (TI) for Astronautics (AERSP 309) 

Fall 2012

Worked with a faculty member (Dr. Melton) for about 10 hours per week grading, meeting students during office hours, conducting help and review sessions, writing homework and exam questions, and giving short lectures

o

Mathematics and Language Tutor for Penn State Learning

o

Learning Assistant for the PSU Dept. of Mathematics

Fall 2010 – Present Spring 2011 – Present

Research Interest Currently working with research advisor (Dr. Spencer) on the determination of the optimal interplanetary trajectory of an arbitrary spacecraft in order to correctly target the moons of Mars. This research includes: 

Interplanetary transfer maneuvers from Earth to Mars and determination of the orbital parameters of the transfer orbits and the relative required ΔV’s and times of flights



Orbit determination of the moons of Mars in order to land/rendezvous with them

Additionally, I am interested in continuing my studies on astrodynamics at the PhD level.

Relevant Design Work Honors Flight Vehicle Design and Fabrication (Lecture and Laboratory) 

Worked on the design and fabrication of the PSU Zephyrus, a full-scale, 22.5-meter wing span Human Powered Aircraft (HPA) designed to compete in the Kremer Prize Competition



Successfully flight tested the aircraft



Worked in teams for the design, fabrication, and integration of parts of the aircraft including: wing, fuselage, propulsion system, connections



Worked with composite materials such as carbon fiber and fiber glass



Have taken this multi-semester course for about 4 years building a deep understanding of systems engineering, hands-on experience, and creativity in problem solving

Spacecraft Mission Design (2 courses, and participation in RASC-AL 2013): 

Worked on a team-based mission proposal for a Human-Focused Mars Mission for the 2013 Revolutionary Aerospace Systems Concepts Academic Linkage (RASC-AL) competition sponsored by NASA; the team participated to the RASC-AL 2013 Competition



Team leader of a group of seven aerospace engineering students (team Martian Lion) with a focus on the Guidance, Navigation, and Control (GNC) subsystem